Strong solutions and stability for a thin-film equation of shear-thinning fluids with contact line in partial wetting

TL;DR

This paper proves the existence and stability of strong solutions for a shear-thinning thin-film equation near contact lines, using variational methods and perturbation analysis, highlighting physical implications for contact-line dynamics.

Contribution

It introduces a novel analysis of strong solutions with nonzero contact angles for shear-thinning fluids, employing a variational approach and stability analysis near the contact line.

Findings

Proved existence of strong solutions perturbing a linear profile.

Established asymptotic stability of these solutions.

Controlled contact-line velocity and analyzed singular behavior.

Abstract

We consider a power-law thin-film equation for strongly shear-thinning fluids. Weak solutions to this equation have been constructed more than twenty years ago by Ansini and Giacomelli. Here, we pass over to analyzing strong solutions with nonzero contact angle (partial-wetting regime), and place emphasis on studying the behavior of solutions near points where the film height vanishes (the contact-line region) by considering perturbations of a linear profile. The leading-order equation in von-Mises coordinates shows similarities with the evolution equation for the $p$-Laplace, though being of fourth order. Using a time discretization, we reduce the leading-order problem to finding a variational solution, and pass to the limit in the discretization scheme on suitably estimating higher-order nonlinear terms in conjunction with compactness arguments. This proves existence and asymptotic stability of strong solutions that are perturbations of the linear profile, and yields control on the contact-line velocity on carefully tracking singular terms in our estimates. While we believe that the transformed equation shows mathematical features the analysis of which stands on its own merit, it also physically corroborates shear thinning behavior as an alternative in resolving the no-slip paradox, as opposed to more standard approaches like introducing slip at the liquid-solid interface.